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% CS395T Data Mining: A Mathematical Perspective
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\lecturenum{8}
\lecturedate{Nov 12, 2008}
\lecturer{Inderjit Dhillon}
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\keywords{Eigenvalue Decomposition, Iterative Methods}
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\begin{enumerate}
\item Problem 25.1, 27.2, 30.7
\item Generate a random $50\times 50$ tridiagonal matrix in Matlab, after setting the seed to be 1 i.e. type ``rand('seed',1);''. Compute its eigenvalues using the ``eig'' command in Matlab. Consider the smallest eigenvalue in magnitude and use following strategies to compute the corresponding eigenvector:
\begin{enumerate}
\item Fix the first component of the eigenvector to be $1$ and then solve for the remaining components of the eigenvector.
\item Fix the last component of the eigenvector to be $1$ and then solve for the remaining components of the eigenvector.
\end{enumerate}
Compare the obtained eigenvector by each of the above strategies with the one generated by Matlab. Is any of the eigenvectors obtained by the above given strategies close to the actual eigenvector (one generated by Matlab)? If not, what can be a possible justification?
\item Let $A$ be $m\times n$ and $B$ be $n\times m$. Show that the matrices $\left[\begin{matrix}AB&0\\B&0\end{matrix}\right]$ and $\left[\begin{matrix}0&0\\B&BA\end{matrix}\right]$ have the same eigenvalues.
\item Give the eigenvalue decompositions and Schur
decompositions of:
\[
A = \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right], \;\;\;\;
B = \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \right], \;\;\;\;
C = \left[ \begin{array}{cc} 5 & -3 \\ -1 & 3 \end{array}\right].
\]
\end{enumerate}
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